Third-party app settings aren't automatically backed up and restored, although many individual applications do this sort of thing themselves, particularly the ones that are really thin clients for some web service. For everything else, like games, the answer is one of the many backup/restore tools, which I believe all require root. I've used Titanium Backup to do this. It lets you make checkpoints of any apps you want, as many times as you want, on your phone. You can then copy those checkpoint files to some other phone and restore from them.